Stephanie,
I am not aware of any special protocol for Papal portraits. They are not ipso facto considered holy images (although I suppose an argument can be made for portraits of Pope St. John Paul II).
If you want to be safe when disposing of them, the customary way to dispose of holy objects is to burn them (or bury them).
As for storing them, the only hint I have is that of storing icons: you never put something on top of an icon other than another icon. We could possibly extend this to images of canonized popes.
Eric
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